The most compact shape depends on the paper

Five rungs of this anchor have measured what a projection does to a reach set’s position, area, width and membership. What none of them has touched is the quantity a reach set is most often reduced to when somebody has to put it in a table: its shape, scored as a single number.

The number is the isoperimetric quotient, and the reach set is the reason it exists. On a plane, $4\pi A / P^2$ is one for a disc and less for everything else, and the disc is the shape that encloses the most area for a given boundary. On a sphere the corresponding statement is Bernstein’s:

$$P^2 \ge 4\pi A - \frac{A^2}{R^2},$$

with equality for a geodesic disc — a cap — of any radius whatsoever, up to and including a hemisphere. So the right score on a sphere is

$$C = \frac{4\pi A - A^2/R^2}{P^2},$$

which is one for every cap and less for everything else, and which the machinery here returns as 0.999994 for a densified 12° disc. That is the isoperimetric equality, exercised rather than cited.

Why the reach anchor owns this #

A compactness score is a statement about a set, which is what this anchor is for, and the shape it is normalised against is the one this anchor’s first rung is about: a circle of a distance is not a circle establishes that the set within a stated travel cost is a geodesic disc on the ground and something else on the page, and every compactness score in use asks how far a region is from being exactly that.

So the score is a reach set used as a ruler. Everything this anchor has measured about how badly a page renders a disc is therefore measured, once more, in the units the score is quoted in — and the answer is not the same, because a score is a ratio and the errors in its numerator and denominator do not cancel.

What everybody actually computes #

Nobody computes $C$. What gets computed is $4\pi A/P^2$ from the geometry in the file, and the geometry in the file is projected.

That is the standing rule of this collection — an operation performed after the map is an operation performed on the map — applied to a quantity that has no coordinates in its output at all. A compactness score is one number between zero and one. There is nothing in it to suggest that a coordinate system was involved.

The scores are used. Polsby–Popper and Reock scores appear in redistricting litigation in the United States and in several state constitutions; catchment compactness appears in hydrology; shape indices appear in landscape ecology and in urban form studies. In every one of those the input is a shapefile and the shapefile is in a projected coordinate reference system.

The disc is not always the champion #

Four projections of ten — the plate carrée, the Lambert azimuthal equal-area, the Robinson and the Miller cylindrical — place a mildly elongated oval above the geodesic disc on the page, at a centre of 45° north.

The mechanism is not subtle once it is stated. Each of those four compresses in one direction at 45° north relative to the other, so a shape that is elongated the other way is squeezed back toward round while the disc is stretched away from it. The oval used here has axes of 13.2° and 10.9°, a ratio of 1.21, and that is roughly the ratio those projections need to undo. The page’s most compact shape is the one whose own elongation cancels the projection’s, and on the ground that shape is not the best one and the difference is 1.4 per cent of the score.

A projection stretching by 1.21 in a stated direction is not an extreme projection. It is what any of these does over a region of twenty-odd degrees at mid-latitude — which is the size of a district, a catchment, or a country.

The test that removes every other explanation #

The oval and the disc are different shapes, so a sceptic can say that comparing them is comparing two things and the projection is only one of the differences. There is a comparison with no such escape.

Rotating a shape about its own centre on a sphere is an isometry. It changes nothing: not the area, not the perimeter, not any intrinsic quantity whatsoever, and the machinery returns identical ground scores to nine decimal places, which is the arithmetic saying so.

On the page the two differ by 4.7 per cent on the plate carrée, and by measurable amounts on every projection here except the ones whose distortion happens to be isotropic at that latitude.

So the score is not a property of the region. It is a property of the region together with its orientation relative to the projection’s principal directions, which are the axes of Tissot’s indicatrix and are decided by the map rather than by the ground. Two districts of identical shape, one running north–south and one north-east, score differently, and the difference is larger than the margins these scores are argued over.

How large the disagreements are #

The ordering is the headline and the sizes matter more for anybody using a score.

Both signs, and no pattern that a single correction factor could absorb. A shape whose long axis runs east–west on a projection that stretches east–west comes out rounder; the same shape turned comes out thinner; and a shape that is already round moves by whatever the projection’s local anisotropy happens to be.

The folded corridor is worth a line on its own. It scores 0.081 on the ground, which is the sort of number a compactness test is deployed to catch, and 0.096 on the page — an 18 per cent improvement in the score, from a map. A threshold set at 0.09 would pass it on one sheet and fail it on another.

That last point deserves its own sentence, because it is the one a reader will reach for. A conformal projection does not fix this either. Conformality preserves angles in the limit of a shrinking figure; a twenty-degree region is not in that limit, and Mercator moves the disc’s score by 0.01 per cent at this centre and much more at higher latitudes, because the indicatrix is a limit and a district is not.

Equal-area does not save it #

The obvious repair — the one that fixes almost every failure of this kind elsewhere in this collection — is to use an equal-area projection. It does not work here and the reason is one line.

$C$ has an area in the numerator and a perimeter squared in the denominator. An equal-area projection preserves the first exactly and does nothing whatever for the second. The Gall–Peters and the Lambert azimuthal equal-area are both in the table above; the first happens to preserve the order of these nine shapes and the second is one of the four that dethrone the disc, and neither outcome has anything to do with their being equal-area.

That is a genuinely different situation from the thematic anchor, where every failure vanishes identically on an equal-area map and the recommendation is unambiguous. There is no projection that preserves boundary length, because a projection that preserved all lengths would be an isometry and no such map exists. A perimeter is not a preservable quantity, so no choice of projection makes a page compactness score right.

There is only one repair and it is the same one as always: compute on the ellipsoid. The area and the geodesic perimeter are both available in every serious geometry library, both are one call, and both remove this failure entirely on any projection whatsoever.

What was computed, and how #

Nine shapes, every one a stated closed form: a geodesic cap, two ellipses of geodesic radii differing only by a 45° rotation, a regular spherical hexagon built from great-circle edges, two latitude–longitude boxes of different aspect, two elongated lenses, and a folded corridor. All centred at 45° north on the central meridian, all densified to 720 boundary points.

The spherical area comes from the excess formula summed edge by edge, which is stable for the very small edges a densified boundary has; the naive interior-angle sum loses every digit there. It is checked against the cap’s closed form $2\pi R^2(1-\cos r)$, and the isoperimetric equality is the assertion: the cap’s score must be one to two parts in ten thousand, and no shape on the ground may exceed one at all. A sign error or a wrong constant in the quotient would break both.

The page quantities are a shoelace and a Euclidean perimeter over the projected vertices. The quotient is scale-free, so no normalisation is needed and none is applied — which matters, because a normalisation is a place a thumb can rest.

The rejecting form of the rung’s claim is two-sided: some projection must dethrone the disc, or there is nothing to report, and some projection must not reorder anything, or the effect is in the sampler rather than in the maps. Both hold.

Where the model stops #

Nine shapes is not a survey. Which projections dethrone the disc depends on which shapes are in the set, and adding a shape can change the answer without changing any of the old scores. The claim is that the page order differs from the ground order, not that any particular projection is worst.

One centre. Everything here is at 45° north. The effect grows toward the poles for the cylindrical projections and shrinks toward the equator, and a study of districts spread across a large country would see all of it at once.

The boundaries are exact. Every vertex here is on the shape it is supposed to be on, which is exactly what a real boundary is not. The area is unbiased and the perimeter is not prices what vertex noise does to the same score, and it is a separate failure with the same denominator: the perimeter’s bias makes the score too low, always, and worse the more finely the boundary is captured.

Reock’s score is not measured here. It is the ratio of a shape’s area to that of its smallest enclosing circle, which has the same page-versus-ground problem in a different form — the smallest enclosing circle on the page is not the image of the smallest enclosing circle on the ground, and a circle of a distance is not a circle is why.

And a compactness score is a poor instrument in the first place. Whether a district is gerrymandered is not a question about its isoperimetric quotient, and every practitioner in that field says so. This rung is about the number’s reliability rather than about its usefulness.

The generalisation #

This anchor’s rule has been the same for six rungs — a set is not a route, and a set’s quantities do not survive a projection the way a route’s position can. The isoperimetric quotient sharpens it in a way none of the earlier rungs could.

Area survives on an equal-area map. Position survives on the right azimuthal. Angle survives on a conformal map. Nothing preserves boundary length, so any statistic with a perimeter in it is unfixable by choosing a projection, and the choice of projection is the only lever most producers have.

That splits this collection’s failures into two kinds, and the split is worth carrying. There are failures where a projection can be chosen to make the quantity exact — and the essays about them end in a recommendation. And there are failures where no projection can, because the quantity involves a length in general position, and those end in the only other repair available: compute on the body, not on the page. A compactness score, a fractal dimension, a shape index and a boundary length are all of the second kind.

Who found it, and when #

The planar isoperimetric inequality is Greek in its statement and nineteenth-century in its proof; the spherical version with the $A^2/R^2$ correction is standard in the differential geometry of surfaces and is credited to Bernstein and to Schmidt in the 1930s.

The compactness scores are twentieth-century political science. Reock’s is from 1961 and Polsby and Popper’s from 1991, and both were proposed as objective, computable substitutes for a judgement about shape. Both are computed from projected geometry as a matter of course, and the projection is stated in almost no published score.

The one place the issue surfaces is in the American redistricting literature, where the choice of projection for computing compactness is occasionally raised and is usually settled by adopting a state plane coordinate system — which is a good decision, since a state plane zone is a conformal projection over a small region and its distortion is parts per million. The failure measured here belongs to studies at continental or global scale, and to any comparison between jurisdictions in different zones.

Why the score’s users are the least able to notice #

The scores in question are not academic instruments. They are used to decide whether a district is acceptable, they appear in expert testimony, and they are compared across jurisdictions and across decades — and every property of that use makes the defect harder to see.

The comparison is between shapes rather than against a standard. Nobody asks whether a district’s Polsby–Popper score is right; they ask whether it is lower than another district’s. A systematic distortion shared by both leaves the ordering intact and the comparison feels sound, which is precisely the situation in which a shared error is invisible.

The numbers are stable. Recomputing a score from the same shapefile gives the same answer to many digits, and stability reads as reliability. What is stable is the arithmetic; the projection under it is a constant of the pipeline rather than a parameter anybody varies.

The users are not cartographers. A political scientist, a court, or a legislature has no reason to ask which plane the areas and perimeters were computed in, and the software does not ask either — it computes on whatever coordinates the file holds.

And the defect crosses exactly the boundary the scores are used across. Within one state plane zone the distortion is parts per million and the scores are fine, which is the good decision the literature already made. The comparisons that matter politically — one state against another, this decade’s map against the one before a datum or projection change — are the ones that leave the zone.

One more property of the use makes it worse, and it is about who checks. A score entering a legal argument is scrutinised by opposing experts, which sounds like the strongest possible review — and both sides compute from the same shapefiles with the same libraries, so a disagreement about the number is nearly impossible and a disagreement about the frame it was computed in has never been raised.

So the practice is safe where it was examined and unexamined where it is not safe, which is the ordinary shape of a defect that survives. The remedy is the same one this collection keeps arriving at: compute the area and the perimeter on the ellipsoid, where both are defined without reference to any sheet, and report the projection when a projected computation is used anyway.

Where the ladder goes next #

Every score in this essay was computed from a boundary read at 720 points. Nothing in it asked whether 720 was the right number, and the shapes were chosen to be smooth so that it would not matter. A real boundary is not smooth: its length depends on the scale it is drawn at, without limit, which is a fact this collection established a long way from here. The next rung puts the two decisions side by side on one boundary and measures which of them moves the score more. It is not close.